Classification of tiling $C^*$-algebras

Ito, Luke J.; Whittaker, Michael F.; Zacharias, Joachim

doi:10.48550/arxiv.1908.00770

Cited by 3 publications

(6 citation statements)

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“…(2) Corollary 7.8 naturally applies to many minimal almost finite groupoids, such as certain minimal almost finite geometric groupoids introduced in [6] and groupoids of repetitive aperiodic tillings of R d with finite local complexity, e.g., Penrose tillings, considered in [9].…”

Section: Xin Mamentioning

confidence: 99%

“…The main obstruction is that, in a general fiberwise amenable groupoid G, so far there has been no topological tiling results for any compact "Følner" set that is applicable to general groupoids. However, we remark that it was developed a version in [9] that is applicable to certain tilling groupoids. Nevertheless, the methods developed in [13] called extendability and almost elementariness actually are good enough here to construct an c.p.c order-zero map from matrix algebras to the groupoid algebras.…”

Section: Introductionmentioning

confidence: 99%

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Fiberwise amenability of ample étale groupoids

Ma¹

2021

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Let G be a locally compact σ-compact Hausdorff ample groupoid on a compact space. In this paper, we further examine the (ubiquitous) fiberwise amenability introduced by the author and Jianchao Wu for G. We define the corresponding concepts of Følner sequences and Banach densities for G, based on which, we establish a topological groupoid version of the Ornstein-Weiss quasitilling theorem. This leads to the notion of almost finiteness in measure for ample groupoids as a weaker version of Matui's almost finiteness. As applications, we first show that C * r (G) has the uniform property Γ and thus satisfies the Toms-Winter conjecture when G is minimal second countable (topologically) amenable and almost finite in measure. Then we prove that the topological full group [[G]] is always sofic when G is second countable minimal and admits a Følner sequence. This can be used to strengthen one of Matui's result on the commutator subgroup D[[G]] when G is almost finite. Concrete examples are provided.

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Section: Xin Mamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fiberwise amenability of ample étale groupoids

Ma¹

2021

Preprint

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“…Example 9.13. Recently, in [IWZ19], Ito, Whittaker and Zacharias established Zstability of Kellendonk's C * -algebra of an aperiodic and repetitive tiling with finite local complexity through generalizing the approach for group actions in [Ker20] to groupoid actions. In addition, they showed that such a C * -algebra is a reduced C * -algebra of a locally compact Hausdorff étale second countable minimal principle almost finite tiling groupoid.…”

Section: Then We Can Do the Same Decomposition Process For All Multis...mentioning

confidence: 99%

“…Inspired by the Følner set approach to (group-theoretic) amenability, Matui [Mat12] introduced a more general notion called almost finite groupoids 1 , which, like AF groupoids, also applies to ample étale groupoids, but it only demands that every compact subset of a topological groupoid is almost contained in an elementary subgroupoid in a Følner-like sense. Almost finiteness strikes a remarkable balance between applicability and utility: it was shown to be enjoyed by transformation groupoids arising from free actions on the Cantor set by Z n (later generalized in [KS20]; see below), as well as those arising from aperiodic tilings ( [IWZ19]); on the other hand, this notion has found applications in the homology theory of ample groupoids, topological full groups, and the structure theory and K-theory of reduced groupoid C * -algebras, and deep connections to an increasing number of other important properties have been established (see, for example, [Mat15,Mat17,Nek19,Ker20,Suz20]).…”

Section: Introductionmentioning

confidence: 99%

“…Kerr's method appears difficult to generalize, for it makes use of the fact that the levels in the Rokhlin towers witnessing almost finiteness are labeled by group elements, which allows one to apply Ornstein and Weiss' theory of quasi-tilings to carefully manipulate the towers. Ito, Whittaker and Zacharias [IWZ19] managed to extend Kerr's method and result to the case of étale groupoids from aperiodic tilings, exploiting the fact that étale groupoids from aperiodic tilings are reductions of transformation groupoids associated to R nactions. Nevertheless, this method appears unsuitable for groupoids without obvious underlying group structures.…”

Section: Introductionmentioning

confidence: 99%

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Almost elementariness and fiberwise amenability for étale groupoids

Ma¹,

Wu²

2020

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In this paper, we introduce two new types of approximation properties for étale groupoids, almost elementariness and (ubiquitous) fiberwise amenability, inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show that, in their respective scopes of applicability, both notions of almost finiteness are equivalent to the conjunction of our two properties. Our new properties stem from viewing étale groupoids as coarse geometric objects in the spirit of geometric group theory. Fiberwise amenability is a coarse geometric property of étale groupoids that is closely related to the existence of invariant measures on unit spaces and corresponds to the amenability of the acting group in a transformation groupoid. Almost elementariness may be viewed as a better dynamical analogue of the regularity properties of C * -algebras than almost finiteness, since, unlike the latter, the former may also be applied to the purely infinite case. To support this analogy, we show almost elementary minimal groupoids give rise to tracially Z-stable reduced groupoid C * -algebras. In particular, the C * -algebras of minimal second countable amenable almost finite groupoids in Matui's sense are Z-stable. Contents 1. Introduction 1 2. Preliminaries 8 3. Amenability of extended coarse spaces 11 4. Coarse geometry of étale groupoids 15 5. Fiberwise amenability 21 6. Almost elementary étale groupoids and groupoid strict comparison 30 7. Almost elementariness and almost finiteness 39 8. Small boundary property and a nesting form of almost elementariness 46 9. Tracial Z-stability 56 References 67This is one reason why we decided not to use the word "finite" in the name of our new notion, after a helpful suggestion of George Elliott to the second author.

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Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

Li¹

2022

Preprint

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Inspired by work of Szymik and Wahl on the homology of Higman-Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui's AH-conjecture for all minimal, ample groupoids with comparison.

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Classification of tiling $C^*$-algebras

Cited by 3 publications

References 39 publications

Fiberwise amenability of ample étale groupoids

Fiberwise amenability of ample étale groupoids

Almost elementariness and fiberwise amenability for étale groupoids

Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

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